A hybrid computational scheme for singularly perturbed Burgers’-Huxley equation

This paper aims to construct and analyze a hybrid computational method for the nonlinear singularly perturbed Burgers’-Huxley equation. The presence of the perturbation parameter and non-linearity in the considered problem makes it difficult to solve the problem analytically and using classical numerical techniques on uniform step sizes as ε goes small. To elucidate such limitations, one can rely on non-classical numerical techniques. In this paper, one such parameter uniform numerical method is designed for the considered problem. The method is constructed: • Firstly, the non-linear terms are linearized via Newton-Raphson-Kantorovich technique. The linearized problem is discretized by the implicit Euler method in the temporal direction.• Secondly, the obtained equation is solved by employing the hybrid computational method comprised of the cubic spline in tension method in the inner layer region and the midpoint upwind method in the outer layer region on a piecewise uniform Shishkin mesh.• Finally, an error analysis of the method is done and observed that the proposed method is parameter uniform convergent with the order of convergence O(τ+N−2ln3N). Three examples are presented and the results are compared to some existing schemes in the literature to demonstrate the reliability of the proposed scheme.

, bifurcation and chaos [23] , membrane models based on the dynamics of potassium and sodium ion fluxes [16] and references therein.To investigate the behaviour of the physical phenomena of these problems, it is often vital to find their solution numerically.However, it is very tough to solve non-linear SPBHE analytically due to the presence of the perturbation parameter and the nonlinearity in the problem.As  tends to zero, the classical computational techniques are fail to provide acceptable solutions for such types of problems.So, it is obligatory to develop robust numerical methods that solve the problems under consideration effectively.
In this work, we are motivated to develop and analyse an  − uniformly convergent numerical scheme for SPBHE.Developing parameter uniform numerical techniques for SPBHE is a desirable task and active for research.A detailed description of the formulation of the scheme and analysis of the scheme is given in the subsequent sections.
Notations Throughout the paper, ‖⋅‖ ∞ is defined by ‖‖ ∞ = sup ( ,  )|( ,  )| denotes standard supremum norm for a function  defined on some domain . and  denote the number of mesh points in space (  ) and time (  ) directions, respectively. denotes a generic positive constant independent of the  and the mesh sizes.

Spatial discretization
On The mesh [0,1] is given by: for  = 0(1) ∕2 , where Now, we fully discretized the problem (10) via a hybrid computational scheme which is based on the midpoint upwind method and cubic spline in tension method in the outside and the inside layer region, respectively.
Eq. ( 11) can be written in the form of systems of equations as: where ,

MethodsX 12 (2024) 102574
Error analysis  − uniform convergence analysis of the proposed numerical scheme is the primary focus of this section.We analyzed the  − uniform convergence of the proposed method divided into two cases as follows.

Numerical examples and results
Three examples are presented in this section to ensure that the proposed scheme is applicable and efficient.In all cases, we did numerical experimentations by using  1 = 1  − 02 and  2 = 4 .9  − 01 .Since the exact solutions of these examples do not exist, we use the double mesh principle to compute the maximum point-wise absolute errors [3] : Example 1.8.Consider the following SPBHE: Example 1.9.Consider the following SPBHE:  and ∧ , between the proposed scheme and the scheme in [17] at various values of  ,  and  for Examples 1.7 and 8 , respectively.These show the accuracy of the proposed scheme. ,  ,  , , ∧ ,  and ∧ , comparison between the proposed scheme and the scheme in [13] at various values of  ,  and  for Example 1.9 is tabulated in Table 3 .It can be observed that the proposed scheme has more accurate results than results in [13] .

Conclusions
Herein, the non-linear singularly perturbed Burgers'-Huxley problem was studied via a hybrid computational scheme.The numerical scheme is based on discretizing the differential equation using the implicit Euler method for the time variable and the hybrid numerical scheme composition of the midpoint Euler method in the outer layer and the cubic spline in compression method in the inner layer on a piecewise uniform Shishkin mesh for the space variable.The scheme is analyzed for  − uniform convergence.Several examples are discussed and compared to some existing schemes in the literature to prove the effectiveness of the presented scheme.
In each example, we calculated the  ,  ,  , and the corresponding ∧ ,  and ∧ , for different values of  and .From the results in Tables 1-3 , we observe that the maximum point-wise error decreases as  and  increase for each value of  goes small.We observe that the maximum point-wise errors are stable as  goes small for each  and  and which depicts that the method is convergence with independent of  .Our findings have confirmed that the proposed method reveals high accuracy and can yield results that are on par with, or even superior to, some available numerical schemes for tackling the SPBHEs.This achievement emphasizes the potential of the proposed scheme as a powerful tool for addressing not only the SPBHEs but also other important nonlinear partial differential equations that come across in numerous engineering and scientific contexts.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Figs. 1
Figs. 1-3 display the numerical solutions for Examples 1.7 -9 .These figures show as  → 0 the width of the boundary layer increases and a strong boundary layer is created near  = 1 .Fig.4shows the log-log plot of the maximum absolute errors.It can be seen that the error decreases as  increases.

Fig. 4 .
Fig. 4. The maximum absolute error via loglog plot at various values of  for Examples 1.7 and 1.8 .